What is the net-change formula for a continuously differentiable function on an interval?

Continuity ensures that as the interval width $h$ approaches zero, the average value of $f$ over $[x, x+h]$ converges to the instantaneous value $f(x)$. Without continuity, the local behavior might oscillate wildly or have jumps, preventing the limit of the difference quotient from settling on a single well-defined value $f(x)$.

Conditions: $A(x) = \int_a^x f(t) dt$; $f$ is continuous at $x$

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

The notation $df/dx$ motivates thinking of tiny vertical changes paired with tiny horizontal changes, and informally canceling $dx$ helps intuition. However, it is misleading because the derivative is defined by a limit, not by dividing two ordinary numbers.

Conditions: Interpreting the differential notation in the context of the theorem.; Understanding the formal definition of the derivative.

The geometric meaning is that each small horizontal step uses its own local slope. The resulting vertical increments accumulate to the total rise from $f(a)$ to $f(b)$.

Conditions: Viewing the theorem through the lens of Riemann sums or local linear approximation.; The function is differentiable on the interval.

The accumulation function $A(x)$ defines area by integrating from a fixed lower endpoint to a variable upper endpoint $x$. For $x \ge 0$, this represents ordinary geometric area.

Conditions: The lower endpoint is fixed.; The upper endpoint $x$ is variable.; Signed integration is used to handle direction and negative values.